Solow's Model

1. One of them is sollow's equation, change in capital stock ?

A.dk=i-$k

B.dk=sf(k*)-$k

C.sf(k*)=$k*

D.All of the above✔

2. Under sollow's model technical progress :-

A.f (k,i)

B.f (k,L)

C.f(k,l×e)✔

D.None

3. What is effective worker ?

A.l×i

B.l×s

C.l×e✔

D.All of the above

4.Technical progress means :-

A.sf (k)-($+n+g)✔

B.f (k*)-($+n+g)k*

C.Both

D.None

Some tips for sallow model based problem :-

Given: The production function is Y=K1/2L1/2, households save 20% of their income and capital lasts for 20 years.This continues our earlier example. From last time, we found that f(k)=k1/2, s=0.2 and δ=0.05. We also found that the steady-state capital stock level was k=16. At k=16, we found that y=f(k)=4, c=3.2 and i=0.8.

1. Calculate c* (consumption in the steady-state) for k*=90, 100, and 110. (recall that in the steady-state, sf(k)=δk, so c*=f(k)-δk)

c=4.99, 5.00, 4.99 for k*=90, 100, and 110 respectively

2. For the above production function, the MPK is

0.5k -1/2 (which is the derivative of f(k) with respect to k). What is the Golden Rule steady-state capital stock level, k*gold?

The Golden Rule condition is that the marginal product of capital is equal to depreciation:

MPK=δ

1/(2k1/2)=0.05

solving for k gives k*gold=100

This is the specific capital stock that maximizes consumption

3. Must the saving rate, s, be increased or decreased to move from the present steady-state level, k*=16, to k*gold?

Since we are starting at k*=16, and k*gold=100, we need to increase s to build the capital stock.

4. What must s become so that k*=k*gold? (Hint: recall the condition for the steady-state, sf(k)=δk)

s k1/2=0.05k

substitute in k=100, and

s(1001/2)=0.05(100)

thus s=0.50=50%

5. Say we start at k=16, but that s changes to the level in question (4). In this first period of transition, what is c, i, and Δk?

In other words, we are starting at the initial capital stock level, but raise the saving rate which will eventually achieve k*gold and maximum consumption.

c=(1-s)y=(1-s)f(k) =(1-s) k1/2

c=(1-.5)(16)1/2

c=2

Notice that consumption is much lower than initially (c=3.2), because people have increased saving rates.

investment=total savings

i=sf(k)=0.5( k1/2)= 0.5(16)1/2

i=2

Δk =sf(k)- δk=0.5(16)1/2-0.05(16)

Δk=1.2

Thus, in the first period, the capital stock rises by 1.2.

6. In the second period of transition, what is k, y, c, and i?

In the second period, the capital stock starts at 17.2(=16+1.2)

k=17.2

y= f(k)=k1/2= (17.2)1/2= 4.15

c=(1-s)y=(1-s)f(k) =(1-s) k1/2=(1-.5) (17.2)1/2=2.08

i=sf(k)=0.5( k1/2)= 0.5(17.2)1/2=2.08

notice that consumption in the second period is above the first period.

7. What will c be at the Golden Rule level of k?

Consumption at k=100 is c=(1-s)y=(1-s)f(k) =(1-s) k1/2=(1-.5) (100)1/2 =5